#9017285

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#41655	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,5)(4,9)(6,16)(7,10)(8,14)(11,13)(12,15)(17,18,22,24)(19,25,27,21)(20,23,30,28)(26,31,32,29);(1,3,7,6)(2,5,10,16)(4,12,14,11)(8,13,9,15)(17,19)(18,21)(20,26)(22,27)(23,29)(24,25)(28,31)(30,32)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(1,4,7,14)(2,8,10,9)(3,11,6,12)(5,15,16,13)(17,18,22,24)(19,21,27,25)(20,28,30,23)(26,31,32,29)(33,35,39,46)(34,40,43,44)(36,37,42,45)(38,47,48,41);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,10)(13,16)(17,18)(19,21)(20,28)(22,24)(23,30)(25,27)(26,31)(29,32)(33,36,39,42)(34,38,43,48)(35,45,46,37)(40,41,44,47);(1,5)(2,3)(4,13)(6,10)(7,16)(8,12)(9,11)(14,15)(17,20,22,30)(18,23,24,28)(19,26,27,32)(21,29,25,31)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,6,7,3)(2,5,10,16)(4,11,14,12)(8,13,9,15)(17,21)(18,19)(20,29)(22,25)(23,26)(24,27)(28,32)(30,31)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924481);